1. A triangulation of a polygon is a way to divide the polygon into triangles by adding diagonals that do not

intersect each other (except possibly at the vertices of the polygon). How many diagonals does one need to add

in order to triangulate a regular 2021-gon?

2. Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 40.

Which of a, b, c, d, or e can be found? (ITS NOT 6+7+8+9+10, it can only be one of the letters.)

3.A chef places four tablespoons of water into an eight-tablespoon measuring cup and four tablespoons of vinegar

into a second cup of the same size. She then pours half the water from the first cup to the second and, after

stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first

cup is now vinegar?

Guest Jan 30, 2021

edited by
Guest
Jan 30, 2021

#1**+1 **

1. From the formula, a regular 2021-gon has 2021*2018/2 = 2039189 diagonals.

2. You can find a, becuase a is part of the formula for the sum

a + (a + d) + (a + 2d) + ... + (a + (n - 1)d) = (2a + (n - 1)d)/2.

Guest Jan 30, 2021

#2**+1 **

2. Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 40.

Which of a, b, c, d, or e can be found?

c, the middle term, is the one that can be found

We have an odd number of terms..so this will = the sum of the terms / the number of terms = 40 / 5 = 8

To see this...suppose we have

1 + 4 + 7 + 10 + 13 = 35

The middle term is 35/5 = 7

CPhill Jan 30, 2021

#3**+1 **

3.A chef places four tablespoons of water into an eight-tablespoon measuring cup and four tablespoons of vinegar

into a second cup of the same size. She then pours half the water from the first cup to the second and, after

stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first

cup is now vinegar?

After pouring, the first cup contains 2 Tsp of water

And the mixture in the second cup is 2 parts water and 4 parts vinegar = 2 Tsp of water and 4 Tsp of vinegar

The mxing does not affect the ratio

So..... she pours 1/2 of this back into the first cup = 1 Tsp of water and 2 Tsp of vinegar

And the first cup now contains 3 Tsp of water and 2 Tsp of vinegar

So....the first cup contains 2 / ( 2 + 3) = 2/5 vinegar

CPhill Jan 30, 2021